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On their 20th birthday, identical twin
astronauts volunteer for an experiment.
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Terra will remain on Earth, while Stella
will board a spaceship.
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Stella’s ship will travel at 86.6% the
speed of light
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to visit a star that is 10
light-years away,
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then return to Earth at the same speed.
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As they prepare to part ways,
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the twins wonder what will happen
when they’re reunited.
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Since a light year is exactly the distance
light can travel in a year,
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Stella’s journey should take 23 years.
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But from having studied
special relativity,
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the twins know it’s not that simple.
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First of all, the faster an object moves
through space,
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the slower it moves through time
compared to an unmoving observer.
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This relationship can be quantified with
something called the Lorentz factor,
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which is defined by this equation.
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And secondly, the length of a moving
object as measured by an observer at rest
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will contract by the same factor.
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At 86.6% of the speed of light
the Lorentz factor is 2,
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meaning time will pass twice as slowly
aboard the spaceship.
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Of course, Stella won’t notice
time slowing down.
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That’s because all time-based processes
in the ship will slow down as well–
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clocks and electrical devices;
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Stella’s biological activities including
her rate of aging
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and her perception of time itself.
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The only people who could notice time
on the moving spaceship
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passing slower for Stella
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would be observers in an inertial,
or non-accelerating, reference frame–
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like Terra back on Earth.
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Thus, Terra concludes that when they meet
back on Earth,
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she’ll be older than Stella.
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But that’s just one way of
looking at things.
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Because all movement is relative,
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Stella argues it would be just as valid to
say her spaceship will stand still
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while the rest of the universe,
including Terra, moves around her.
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And in that case, time will pass twice as
slowly for Terra,
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making Stella the older twin in the end.
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They can’t each be older than the other,
so which one of them is right?
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This apparent contradiction is known as
the “Twin Paradox.”
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But it’s not really a paradox–
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just an example of how special relativity
can be easily misunderstood.
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To test their theories in real-time,
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each of the twins agrees to send
a burst of light to the other
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every time a year has passed for them.
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Unlike other objects, the speed of light
is always constant
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regardless of an observer’s
reference frame.
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A light burst sent from Earth will be
measured at the same speed
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as a light burst sent from the spaceship,
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regardless of whether it’s on its
outbound or return trip.
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So when one twin observes
a burst of light,
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they’re measuring how long it took the
other twin to experience a year passing,
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plus how long it took for light
to travel between them.
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We can track what’s happening on a graph.
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The X axis marks distance from Earth,
and the Y axis tracks the passage of time.
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From Terra’s perspective, her path will
simply be a vertical line,
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with distance equal to zero
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and each tick on the line equivalent
to a year as she perceives it.
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Stella’s path will stretch from the same
origin to a point 11.5 years in time
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and 10 light-years in distance from Terra…
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before converging again at zero
distance and 23 years’ time.
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At her first one-year mark,
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Terra will send a pulse of light from
Earth towards Stella’s spaceship.
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Since light takes a year to travel
one light-year,
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its path will be a 45-degree
diagonal line.
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And because Stella is
traveling away from it,
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by the time the light catches up to her,
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over 7 total years will have passed for T
erra, and over 4 for Stella.
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By the time Stella observes
Terra’s second burst,
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she will already be on her return journey.
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But now, since she’s moving towards the
source of the light,
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it will take less time to reach her,
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and she’ll observe the bursts
more frequently.
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This means that Stella observes Terra
aging slowly
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for the first half of her journey,
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but aging rapidly during the return half.
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Meanwhile for Stella, it seems as though
Terra, the destination star,
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and the whole universe are
moving around her.
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And because of length contraction,
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Stella observes the distance between
them shrinking by a factor of 2.
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This means each leg of the trip will only
take about six years
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from Stella’s perspective.
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When she sends the first signal to Earth,
two years will have passed for Terra.
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Stella will send four more light bursts
during her outbound journey,
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each one from farther away.
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By the time Terra observes the first pulse
from Stella's inbound journey,
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over 21 years will have passed for her.
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For the rest of Stella's return home,
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Terra receives multiple light
bursts each year.
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Thus, Terra observes Stella aging slowly
for about 90% of their 23 years apart,
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and aging rapidly during the last 10%.
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This asymmetry accounts for why the
paradox isn’t really a paradox.
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Although each twin witnesses time
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both speeding up and slowing
down for the other,
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Stella sees an even split,
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while Terra sees Stella aging slowly for
most of the time they’re apart.
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This is consistent with each twin’s
measurement of the space voyage,
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which takes 23 Earth years, but only
11.5 as experienced aboard the ship.
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When the twins are reunited, Terra will be
43 years old, while Stella will be 31.
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Where Stella went wrong
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was her assumption that she and Terra had
equal claim to being inertial observers.
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To be an inertial observer, one has to
maintain a constant speed and direction
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relative to the rest of the universe.
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Terra was at rest the entire time,
so her velocity was a constant zero.
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But when Stella changed her direction
for the return journey,
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she entered a different reference frame
from the one she’d started in.
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Terra and Stella now both have a better
understanding of how spacetime works.
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And as twins who are eleven
years apart in age,
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they’re a perfect example
of special relativity.